Cottonwood Pass is preferred because it only climbs to 11000' before descending. Cottonwood Lakes climbs to 12,400' over New Army Pass, so an extra 1,400' vertical, which is significant when carrying a pack with many days' supplies.
You do not need a special permit to climb Mt Whitney from the west and then descend to the west. In prior years, selecting a "Visiting Mt Whitney" option implied you would be traveling through the Inyo National Forest's "Mt Whitney Zone", which only begins on the east side of the Mt Whitney crest. You wouldn't be doing that if you are hiking northbound on the JMT. The Visiting Mt W just increased the fee you paid, and that helped Inyo National Forest administer the Mt. Whitney Zone.
I see Inyo N.F. and Rec.gov have removed the "Visiting Mt Whitney" option. It was there previous years, and JMT hikers would be charged more, even though it technically was not required.